1. Technical Field
This invention relates generally to ultra sensitive silicon sensors (USSS), and more particularly to an electro-thermal feedback circuit providing active thermal isolation using a series temperature-sensing diode configuration.
2. Description of the Background Art
Bolometers are used in many imaging applications because they provide spectral versatility. Bolometers are most commonly used in low cost and reduced sensitivity long-wavelength infrared (LWIR) cameras. Bolometers have also been used, at cryogenic temperatures, for microwave imaging. Bolometers inherently respond over a broad band and this makes them viable for imaging in different spectral bands.
Quantum detectors, operating above liquid Helium temperatures, offer better sensitivity and faster response than bolometers but are not spectrally versatile. The lack of spectral versatility with quantum detectors is particularly evident at mm-waves where no viable quantum detector technology exists. By contrast, in the visible spectrum, high performance solid state quantum detector cameras are commercially available. These imaging cameras are made in silicon and provide high performance e at low cost. Generally, cameras based on quantum detectors provide very impressive imagery at higher photon energies (0.5-2 eV). At lower photon energies (<0.5 eV), or longer wavelengths, quantum detectors require more expensive crystalline materials and complicated integrated circuit (IC) processing. Moreover, at mid-wavelength infrared (MWIR), or longer wavelengths, cryogenic cooling is needed to realize their full performance potential. Hence, quantum detectors have been limited to imaging at photon energies greater than about 0.1 eV, i.e., LWIR. Additionally, the requirement for cryogenic cooling of IR quantum detector is a constant detractor in terms of cost, weight and reliability.
The aforementioned issues with quantum LWIR cameras are the main impetus for searching for alternative approaches. In principle, bolometers with theoretically limited sensitivity are a commercially attractive competitor to quantum based LWIR cameras. Hence, large development efforts were started to improve the sensitivity of bolometers. The expectation is that with theoretically limited sensitivity, there are many applications where LWIR bolometers will offer a viable replacement to LWIR quantum detectors; with the advantages of room temperature operation, adequate sensitivity, good reliability, low cost, and low weight. Additionally, bolometers with theoretically limited sensitivity open the opportunities for room temperature mm-wave imaging cameras.
A bolometer's focal plane is made up of pixels assembled into an x-y array with associated readout circuits. A conventional bolometer pixel 9 and its thermal equivalent circuit are shown in FIGS. 1 and 2, respectively. A bolometer pixel 9, as shown in FIG. 1, includes an absorber element (i.e., detector) 10, represented by a rectangle with area AD that is mechanically supported by several low thermal conductance bridges 12, which sits atop of and are anchored to a thermal bath member 14 having a temperature THB. The detector 10 in each pixel acts as a photon absorber and has a temperature T1. From a scene 8, at temperature Ts, photons (i.e., radiation power) hν are incident on detector 10, of pixel 9, and are absorbed and cause a temperature change δT1 from T1. As shown in FIG. 2, the detector's heat capacity is C1, and the thermal conductance of the bridges 12 between the detector 10 and the heat bath 14 is G2. The scene, at temperature Ts, is radiating energy hν at the detector 10, and this is represented in FIG. 2 as a thermal current QR. The detector 10, in addition to being mechanically attached to the heat bath 14 by the bridge's thermal conductance G2, is radiating power QD1 and receiving QS1 from radiation shields (not shown) attached to heat bath 14. The performance of pixel 9 is analyzed by computing the signal and noise in terms of the small signal equivalent circuit of FIG. 2.
The analysis is based on small signal analysis technique developed for electronic circuits. Translation from a thermal to an electrical domain is made by treating temperatures as voltages; thermal conductances as reciprocal electrical resistances; and thermal currents as electrical currents. Specifically, temperatures TS, T1, THB are treated as voltages. Thermal conductance between the heat bath 14 and absorber element 10 is modeled as a reciprocal resistance with conductance G2. The heat capacity of absorber 10 is represented by capacitor C1. The thermal currents are: from the scene QR=σTS4AD/4F2; from the absorber 10 to the outside world QD1=σT14AD; and from the radiation shields to the absorber 10 QS1=σTHB4AD[1-1/4F2], where σ=5.6697×10-8 W-m−2-° K−4 and F is the optic's f-number. Functional differences between expression for QR, QD1, and QS1 are because photon flux arrives through different solid angles, accounted for by the lens's f-number. Additionally, the detector 10 conducts thermal energy, through conductances G2, to the heat bath 14, at temperature THB. Analytically, the thermal balance conditions at the bolometer are represented as:
                                                                                          Q                  R                                -                                  Q                                      D                    ⁢                                                                                  ⁢                    1                                                  +                                  Q                                      S                    ⁢                                                                                  ⁢                    1                                                              =                            ⁢                                                                    ∫                                          T                      HB                                                              T                      1                                                        ⁢                                                                                    G                        2                                            ⁡                                              (                        T                        )                                                              ⁢                                                                                  ⁢                                          ⅆ                      T                                                                      +                                                      ∫                                          T                      1                                                                                      T                        1                                            +                                              δ                        ⁢                                                                                                  ⁢                                                  T                          1                                                                                                      ⁢                                      jω                    ⁢                                                                                  ⁢                                                                  C                        1                                            ⁡                                              (                        T                        )                                                              ⁢                                                                                  ⁢                                          ⅆ                      T                                                                                                                                              =                            ⁢                                                ∑                                      n                    =                    0                                    ∞                                ⁢                                                                  ⁢                                  [                                                                                                              ∂                          n                                                ⁢                                                                              G                            2                                                    ⁡                                                      (                            T                            )                                                                                                                      ∂                                                  T                          n                                                                                      ⁢                                          ❘                                              T                        HB                                                              ⁢                                                                                                                        (                                                                                          T                                1                                                            -                                                              T                                HB                                                                                      )                                                                                n                            +                            1                                                                                                                                (                                                          n                              +                              1                                                        )                                                    !                                                                    +                                                                                                                                                            ⁢                                                jω                  ⁢                                                                                    ∂                        n                                            ⁢                                                                        C                          1                                                ⁡                                                  (                          T                          )                                                                                                            ∂                                              T                        n                                                                                            ⁢                                  ❘                                      T                    1                                                  ⁢                                                                            (                                              δ                        ⁢                                                                                                  ⁢                                                  T                          1                                                                    )                                                              n                      +                      1                                                                                                  (                                              n                        +                        1                                            )                                        !                                                              ]                                                          (        1        )            
The temperature dependence of G2(T) and C1(T) are included in Equation (1).
Using small signal analysis on a conventional bolometer, shown in FIG. 1, it is reasonable to assume that the derivatives of G1(T) and C1(T) are a weak function of temperature. And for simplicity, only first order terms are retained. At equilibrium, or constant flux conditions, the bolometer's equilibrium temperature T10 is obtained from Equation (1), for ω=0, and is given by:
                              T          1          0                =                              T            HB                    +                                                    Q                R                            +                              Q                                  S                  ⁢                                                                          ⁢                  1                                            -                              Q                                  D                  ⁢                                                                          ⁢                  1                                                                                    G                2                            +                              0.5                ⁢                                                      ∂                                          G                      2                                                                            ∂                    T                                                  ⁢                                  (                                                            T                      1                      0                                        -                                          T                      HB                                                        )                                                                                        (        2        )            
Thus, at equilibrium, the temperature of absorber 10 is different from the heat bath temperature by the net bridge heat current divided by conductance G2*. As expected, the more power the absorber 10 receives, which is equal to (QR+QS1−QD1), the higher will be its operating temperature. Since the radiation shields are attached to the heat bath 14, their temperature [and QS1] is fixed to the heat bath's temperature. Thus the temperature of absorber 10 will change monotonically with changes in scene temperature TS, and these temperature changes are maximized with minimum conductance G2*.
Under dynamic conditions, changes in the scene's temperature by δTS will produce changes in the absorber's temperature equal to δT1, about the thermal equilibrium temperature T10. Assuming the radiation shield is at a constant temperature (i.e., δTHB=δTS1=0), the expression relating δT1 to δTS is computed by taking the differentials of Equation (1) and Equation (2) and combining them. The relationship between δT1 and δTS is given by:
                              δ          ⁢                                          ⁢                      T            1                          =                              (                                          G                R                                                              G                                      D                    ⁢                                                                                  ⁢                    1                                                  +                                  G                  2                  *                                                      )                    ⁢                                    δ              ⁢                                                          ⁢                              T                S                                                    1              +                              jω                ⁡                                  [                                                            C                      1                      *                                        /                                          (                                                                        G                                                      D                            ⁢                                                                                                                  ⁢                            1                                                                          +                                                  G                          2                          *                                                                    )                                                        ]                                                                                        (        3        )            
For brevity, G2*=G2+(T10-THB)∂G2/∂T1 is used to account for the temperature dependence of G2(T). GR=∂QR/∂TS=σTS3AD/F2 is the thermal conductance between the scene and absorber, and GD1=∂QD1/∂T1=4σT13AD is the conductance between the absorber and its surroundings.
According to Equation (3), a δTS change in the scene's temperature causes a δT1 change in the absorber's temperature. Although signal δT1 is monotonically related to δTS, it is attenuated by the factor GR/[GD1+G2*], and the changes are retarded by thermal time constant or radial frequency ωTM=[GD1+G2*]/C1*. The signal is attenuated because: (a) the absorber reradiates at a wider solid angle that it received radiation from the scene [GD1>GR], and (b) a portion of the energy received from the scene is shunted to the heat bath 14 through conductance G2*. For maximum signal, conductances GD1 and G2* should be made as small as possible, and approach GR.
Hence, much effort has been invested into minimizing the thermal conductance G2*. Bridge materials with poor thermal conductivity have been developed, but there are practical limits The time thermal constant responsible for AC attenuation is minimized by minimizing C1*/[GD1+G2*], and this is accomplished by making C1* the absorber's mass very small. However, C1* cannot be made arbitrarily small because it inversely impacts: absorption, pixel fill factor, and thermal noise temperature level. Below, we develop expressions for maximum signal and minimum noise which are given in terms of GD1, G2*, and C1*.
Signal analysis is performed in the temperature domain because LWIR bolometers' signals are based on temperature changes. Correspondingly, noise is also represented as the temperature variance at the absorber 10. Several noise sources contribute to the total temperature variance at the absorber 10, and they are: (1) variance in the scene's photon power absorbed by the absorber 10, δQ2R, (2) variance in the photon power emitted by the absorber 10, δQD12, (3) variance in the radiation shield's photon power absorbed by the absorber 10, δQS12, (4) variance in the thermal bath temperature, δTHB2, and (5) variance in the absorber's temperature produced by noise in readout electronics, δTEL2.
Scene's photon flux fluctuations increase the absorber's temperature variance. Ultimately, the bolometers sensitivity, expressed in noise equivalent temperature resolution (NEΔT), is limited by this temperature variance. The minimum noise level in the scene's signal QR is obtained from the variance in the signal's noise power spectral density δQ2R arriving at the absorber element 10, and it is given by:
                              δ          ⁢                                          ⁢                      Q            R            2                          =                              8            ⁢                                                  ⁢                          A              D                        ⁢            σ            ⁢                                                  ⁢                          k              B                        ⁢                          T              S              5                        ⁢            Δ            ⁢                                                  ⁢            f                                4            ⁢                                                  ⁢                          F              2                                                          (        4        )            
where, Δf represents the absorber's electrical frequency bandwidth, kB is Boltzmann's constant, and σ is the Stefan-Boltzmann constant. The 4F2 factor attenuates the signal flux at the absorber in accordance to the solid angle define by the optics F#. The signal's noise power spectral density δQ2R is readily translated into a temperature variance at the absorber 10, δTS2, by combining Equations (3) and (4) and integrating over frequency. The temperature variance δTS2 at the absorber 10 from the scene's variance δQR2 is given by:
                                                                        δ                ⁢                                                                  ⁢                                  T                  S                  2                                            =                                                2                                      π                    ⁢                                                                                  ⁢                                          F                      2                                                                      ⁢                                                      ∫                    0                    ∞                                    ⁢                                                                                                              A                          D                                                ⁢                        σ                        ⁢                                                                                                  ⁢                                                  k                          B                                                ⁢                                                  T                          S                          5                                                                                                                                                  [                                                                                          G                                                                  D                                  ⁢                                                                                                                                          ⁢                                  1                                                                                            +                                                              G                                2                                *                                                                                      ]                                                    2                                                +                                                                                                            ω                              2                                                        ⁡                                                          [                                                              C                                1                                *                                                            ]                                                                                2                                                                                      ⁢                                                                                  ⁢                                          ⅆ                      ω                                                                                                                                              =                                                                    G                    R                                                        (                                                                  G                                                  D                          ⁢                                                                                                          ⁢                          1                                                                    +                                              G                        2                        *                                                              )                                                  ·                                                                            k                      B                                        ⁢                                          T                      S                      2                                                                            C                    1                    *                                                                                                          (        5        )            
Equation (5) reveals that the temperature variance, induced by the scene 8 on the absorber element 10, is a product of two factors. The first factor is a conductance ratio: the conductance through free space between the scene 8 and the absorber 10, divided by the conductance between the absorber 10 and its surroundings GR/[GD1+G2*]. The second factor is the thermodynamic temperature variance for any object at temperature TS and with a heat capacity C1*. For best performance, the scene's noise should dominate over all other noise sources. This is facilitated with fast optics (small F#) and minimum conductance [GD1+G2*], or excellent thermal isolation. The scene's temperature variance at the absorber 10 is increased by additional sources and these are considered next.
The temperature variance [δT12] at the absorber has several thermal contributors, which include the absorber 10 itself, the heat bath 14, and radiative shields (not shown). In addition, electronic noise indirectly increases the absorber's temperature variance. The largest direct thermal contributor to the absorber's thermal variance is the absorber and this contribution is given as an integral of the power spectral temperature variance density [δT12(ω)] at the absorber by,
                                                                        δ                ⁢                                                                  ⁢                                  T                  1                  2                                            =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                    0                    ∞                                    ⁢                                      δ                    ⁢                                                                                  ⁢                                                                  T                        1                        2                                            ⁡                                              (                        ω                        )                                                              ⁢                                                                                  ⁢                                          ⅆ                      ω                                                                                                                                              =                                                1                                      2                    ⁢                    π                                                  ⁢                                                      ∫                    0                    ∞                                    ⁢                                                                                    4                        ⁢                                                  k                          B                                                ⁢                                                  T                          1                          2                                                ⁢                                                  G                          2                          *                                                                                                                                                  [                                                                                          G                                2                                *                                                            +                                                              G                                                                  D                                  ⁢                                                                                                                                          ⁢                                  1                                                                                                                      ]                                                    2                                                +                                                                              [                                                          ω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                      ]                                                    2                                                                                      ⁢                                                                                  ⁢                                          ⅆ                      ω                                                                                                                                              =                                                                    G                    2                    *                                                                              G                      2                      *                                        +                                          G                                              D                        ⁢                                                                                                  ⁢                        1                                                                                            ⁢                                                                            k                      B                                        ⁢                                          T                      1                      2                                                                            C                    1                    *                                                                                                          (        6        )            
The absorber temperature variance, δT12, is a product of two factors: a leading factor [G2*]/[GD1+G2*] is equal almost to unity. The second factor represents the theoretical temperature variance of an object at temperature T1 and with a heat capacity of C1*. When fluctuations from the lens and radiation shields are included, the leading factor becomes unity. The contributions from the heat bath are made small by increasing the heat bath's heat capacity. Hence, the only remaining contributor to the temperature variance at the absorber is electronic noise.
Electronic noise degrades the bolometer's sensitivity. The electronic noise analysis assumes the most mature and most widely used bolometers: resistive VOx bolometers. Thus, the electronic noise includes the readout voltage noise and noise voltage from the resistive bolometer, biased with a current ICR. The total electronic noise voltage is converted to a temperature variance [δTEL2] by dividing it by [ICR∂RCB/∂TS]2, where ∂RCB/∂TS is the resistive temperature coefficient, ICR is the bias current during readout, and RCB the absorber's resistance. The total electrical noise is a sum of the absorber's and amplifier's noise power spectral densities, and these are represented by white and 1/f noise terms dE2N/df and dE2NA/df, respectively. Integrating these noise terms over radial frequency provides an expression for the equivalent temperature variance produced by the electronic noise voltage given by:
                              δ          ⁢                                          ⁢                      T            EL            2                          =                                            [                                                I                  CR                                ⁢                                                      ∂                                          R                      CB                                                                            ∂                                          T                      S                                                                                  ]                                      -              2                                [                                                                      ⅆ                                      E                    N                    2                                                  /                                  ⅆ                  f                                                            4                ⁢                                                                  ⁢                                  C                  E                                ⁢                                  R                  CB                                                      +                                          ∫                                  ω                  LO                                                  ω                  EL                                            ⁢                                                                    ⅆ                                          E                      NA                      2                                                                            ⅆ                    f                                                  ⁢                                                                  ⁢                                                      ⅆ                    ω                                                        2                    ⁢                    π                                                                                ]                                    (        7        )            
The temperature variance at the absorber due to electronic noise is a sum of two noise terms, multiplied by a factor [ICR∂RCB/∂TS]−2. The two terms are the integrated white noise and 1/f noise integral from the cut-on radial frequency ωLO to the cut-off radial frequency ωEL=1/CERCB. Minimum noise requires that the factor [ICR∂RCB/∂TS]−2 is minimized. Selection of absorbers with a high temperature coefficient ∂RCB/∂TS is important. Second, use of a larger readout current ICR helps, but it creates problems. Larger ICR currents inject I2R power into the absorber that is larger (>1000X) that the signal power resolution (in the pWatts range). The work around large I2R power is to use a pulsed readout to minimize the current heating duty cycle. However, pulsed operation requires wider signal bandwidth, accompanied with more noise. In the analysis it was assumed that the readout amplifiers voltage noise is less than the absorber's electrical noise. Such is facilitated with absorbers that have a large resistance, typically greater than 10KΩ. The ideal white noise power spectral density voltage equals to 12.9 nV/Hz1/2. The 1/f noise power contributions diverge logarithmically with the low cut-on radial frequency ωLO. For the purpose of simplification, 1/f noise contribution is included by increasing the white noise value from 12.9 nV/Hz1/2 to 0.1 μV/Hz1/2. The absorber's total temperature variance δTT2 is approximated by the sum of Equations (5), (6), and (7), and is given by,
                              δ          ⁢                                          ⁢                      T            T            2                          =                              (                                                            k                  B                                ⁢                                  T                  S                  2                                                            C                1                *                                      )                    ⁡                      [                                                            G                  R                                                                      G                                          D                      ⁢                                                                                          ⁢                      1                                                        +                                      G                    2                    *                                                              +                                                T                  1                  2                                                  T                  S                  2                                            +                                                                    C                    1                    *                                                                              k                      B                                        ⁢                                          T                      S                      2                                                                      ⁢                                                      (                                                                  I                        CR                                            ⁢                                                                        ∂                                                      R                            CB                                                                                                    ∂                                                      T                            S                                                                                                                )                                                        -                    2                                                  ⁢                                  0.25                                                            C                      E                                        ⁢                                          R                      CB                                                                      ⁢                                                      ⅆ                                          E                      N                      2                                                                            ⅆ                    f                                                                        ]                                              (        8        )            
The total temperature variance δTT2 reveals how the bolometers sensitivity is affected. The leading factor is the minimum possible thermodynamic temperature variance, governed by scene temperature and the absorber's heat capacity. The second factor shows how this variance is increased by the thermal conductivity between the absorber and its environment relative to thermal conductivity with the scene. This factor reveals that poor thermal isolation lead to sensitivity degradation that cannot be overcome by better electronics or responsivity. This factor is a sum of there terms representing contributions, respectively, from the scene's thermal noise, the absorber's thermal noise, and the electronics' thermal noise. The values of second term is significantly greater than the scene's radiation noise and electronics noise can only further degrade sensitivity, and this is calculated next in terms of NEΔT.
The bolometer's sensitivity (minimum resolvable temperature: NEΔT) is defined at unity signal to noise ratio. The equation for NEΔT is obtained equating the change in absorber's temperature (given by Equation (3)) to the total noise temperature fluctuations at the absorber (given by the square root of Equation (8)). At unity signal to noise ratio, NEΔT equals δTS, making this substitution and solving for NEΔT the following is obtained:
                              NE          ⁢                                          ⁢          Δ          ⁢                                          ⁢          T                =                                                                                                                        k                      B                                        ⁢                                          T                      S                      2                                                                            C                    1                    *                                                              ⁡                              [                                                                                                    G                        2                        *                                            +                                              G                                                  D                          ⁢                                                                                                          ⁢                          1                                                                                                            G                      R                                                        +                                                                                    (                                                                                                            G                              2                              *                                                        +                                                          G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                                                                          G                            R                                                                          )                                            2                                        ⁢                                          (                                                                                                    T                            1                            2                                                                                T                            S                            2                                                                          +                                                                                                            C                              1                              *                                                                                                                      k                                B                                                            ⁢                                                              T                                S                                2                                                                                                              ⁢                                                                                                                                                      ω                                  EL                                                                4                                                            ⁢                                                                                                ⅆ                                                                      E                                    N                                    2                                                                                                  /                                                                  ⅆ                                  f                                                                                                                                                                                    (                                                                                                      I                                    CR                                                                    ⁢                                                                                                            ∂                                                                              R                                        CB                                                                                                              /                                                                          ∂                                                                              T                                        S                                                                                                                                                                            )                                                            2                                                                                                                          )                                                                      ]                                                    1              /              2                                ⁢                                    (                              1                +                                                      ω                    2                                                        ω                    TM                    2                                                              )                                      1              /              2                                                          (        9        )            
This Equation has been condensed by defining ωTM=[GD1+G2*]/C1* and ωEL=1/RECCB. NEΔT is given as a product of three factors. Maximum sensitivity (smallest NEΔT) is achieved by minimizing each one of these factors.
The first factor represents the thermodynamic sensitivity limit dictated by the absorber's heat capacity C1* and scene's temperature TS. The actual sensitivity can only get worse than this theoretical thermodynamic limit
The second in the square brackets is a sum of three terms beginning with [G2*+GD1]/GR, which represents the sensitivity degradation due to thermal loading [GD1+G2*] on the absorber relative to the thermal conductance to the scene GR. This factor is typically greater than 10 and much effort has gone into minimizing it because the sensitivity degradation it produces cannot be undone. The degradation to thermal loading can be minimized with maximizing GR [with fast optics], and minimizing [GD1+G2*] by construing thermal isolation bridges with poor thermal conductivity. Thermal loading has an additional deleterious impact because it also decreases the signal thereby making things more susceptible to electronic readout noise. The second term, in the square brackets, incorporates all the thermal noise contributions from the absorber element [including radiation shields]. Heat bath noise is not included since it can be made arbitrarily small by increasing the heat bath's heat capacity. Thus the second term is much larger than the leading term. The last term is in the square bracket represents readout electronics' noise. Electronics noise can only make things worse and it can be minimized by maximizing the responsivity ∂RCB/∂TS. Optimistically, let's assume that the noise from the resistive bolometer dominates over all electronic noises. For a 104Ω resistive bolometer, the noise dEN2/df is about 2×10−16V2/Hz. 1/f noise will only makes things worse. The bolometer's TCR, equals to ∂RCB/∂TS≈200Ω/° K and lets make G2*/GD1≈10. For: TS≈300° K, AD=0.25×10−4 cm2, and F=1; the third term in the sum in Equation (9) equals to ωEL3×10−10/ICR2. The electronic noise should be significantly less than the second term in the sum, [equals <10]. If (ωEL)=30, the electronics noise becomes insignificant if the circuit current ICR>>0.01 mA, or ICR>0.03 mA. The I2R power associated with ICR>0.03 mA is about 10 μWatts, and this is much larger than the desired power resolution of about 10 pWatts. This large I2R readout power is about a million times larger than the desired sensitivity thereby imposing unreasonable temperature sensitivity. The I2R power can be minimized by minimizing the ICR duty cycle. For a 400×500 elements staring array, the current on time [or I2R power] can be reduced by up to 2×105 times by minimizing the ICR duty cycle. However, a minimum duty cycle maximizes the operating bandwidth given by ωEL, and this increases noise. The larger noise can be attenuated by increasing ICR from 0.03 mA to about 12 mA. Unfortunately, this increases the I2R readout power by about 105, an unacceptable situation. Including 1/f noise makes things even more difficult. Parallel readout can be used to reduce electronic bandwidth, but it will increase the power by the number of parallel channels.
The third factor reveals how sensitivity decreases with frequency ωTM=[GD1+G2*]/C1*. The optimal solution for this factor is to minimize the frequency response ωTM, and this is achieved by making ωTM equal the video frame rate of 30 Hz. These limitations addressed by the active thermal isolation USSS approach described next.
According to Equation (9), thermal loading reduces the NEΔT sensitivity by at least [GD1+G2*]/GR. The responsivity is reduced by [GR/(GD1+G2*)], see Equation (3). The values of GD1 and GR are limited by optics, and the most direct way for improvement is by minimizing the thermal conductivity G2*≈2×10−7 Watts/° K to approach GD1, and GR. Given inherent limitations with material and geometrical approaches, further reductions in G2* required a different approach. The approach, incorporated in the all silicon ultra sensitive silicon sensor (USSS), is to utilize electro-thermal feedback to substantially reduce the effective thermal conductivity of G2*. Electro-thermal feedback offers active, as opposed to passive, thermal isolation and thereby obtain a ten-fold improvement in the thermal isolation. Improved thermal isolation results in increases in photo-responsivity and corresponding improvements in NEΔT. The performance advantages of the USSS approach are detailed in the description of the operation and the structure of a single USSS pixel.
Active thermal isolation, mechanized with electro-thermal feedback, can significantly improve thermal isolation between the absorber element 10 and heat bath 14 in FIG. 1. Thermal isolation improvements with active techniques versus passive techniques directly lead to corresponding improvement in responsivity, and sensitivity. These improvements are shown by analysis of the electro-thermal feedback.
Mechanizing electro-thermal feedback in each USSS pixel requires incorporation of special circuits and components. Specifically, electro-thermal feedback requires: (1) temperature difference sensors, (2) temperature difference amplifier, (3) a heater whose output power dependent on temperature difference, and (4) an approach that incorporates all these into a single pixel.
The USSS pixel approach in FIG. 3 shows a three-temperature-tier design for maximizing area efficiency. The three tier structure includes: an absorber 10 at the top tier at temperature T1, an intermediate stage 16 in the middle at temperature T2, and at the bottom a heat bath 14 at temperature THB. These tiers are mechanically and electrically connected by thermal isolation bridges G1* and G2*.
Electro-thermal feedback requires two temperature sensors to control the power delivered into the intermediate stage 16 so that T2 converges to T1. Conventionally, the temperature difference signal is obtained from two silicon diodes 34 and 40, connected back-to-back (see FIG. 4). Temperature sensing diode 34 is built into the absorber 10 and a temperature sensing diode 40 is built into the intermediate stage 16. Conventionally, temperature sensing diodes 34 and 40 are biased with a constant current and the voltage is a function of temperature. The voltage signal from diodes 34 and 40, biased with a constant current, is about −2 m° K per degree Kelvin.
The voltage signal from the back-to-back diodes 34 and 40 is amplified and controls the temperature T2 of the intermediate stage 16. In each pixel, a single voltage amplifier 18 functions as USSS pixel readout and as a heater with variable output power QH. Combining readout and heater functions facilitates pixel miniaturization. The power QH delivered into the intermediate stage 16 depends on the temperature difference between T1 and T2. If T2>T1 (T2<T1) the quiescent power QH delivered decreases (increases) thereby converging T2 towards T1. The potential across the back-to-back diodes 34 and 40 is amplified and produces an output voltage signal VS. Relative to a dc level, the amplifier output VS will be positive if T2>T1 and negative if T2<T1. As VS increases, the quiescent power QH delivered by the amplifier to the intermediate stage 16 decreases. Conversely, as VS decreases, the quiescent power QH delivered by the amplifier to the intermediate stage 16 increases. The power QH delivered by the amplifier to the intermediate stage 16 is given by,QH=IHB+−IHVs=IHB+−IHAG[ΔΦD1(T1)−ΔΦD2(T2)]  (10)where AG is the amplifier's voltage gain and IH is the dc bias current flowing between B+ and ground.
The amplifier's voltage gain is typically greater than 104, IH is about 1 μA, and B+ is about 10 volts. Since B+ and IH are constants, the amplifier's quiescent power QH heating the intermediate stage varies linearly with the output signal VS, and it becomes a measure of the power received by the absorber. Specifically, as VS decreases (increases) the power delivered by the amplifier increases (decreases). The differential power delivered by the amplifier depends on temperature difference and is readily computed from Equation (10) and given by,
                              δ          ⁢                                          ⁢                      Q            H                          =                                            -                              I                H                                      ⁢                                          A                G                            ⁡                              [                                                                                                    ∂                                                  (                                                                                    ΔΦ                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      ⁡                                                          (                                                              T                                1                                                            )                                                                                )                                                                                            ∂                                                  T                          1                                                                                      ⁢                    δ                    ⁢                                                                                  ⁢                                          T                      1                                                        -                                                                                    ∂                                                  (                                                                                    ΔΦ                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      ⁡                                                          (                                                              T                                2                                                            )                                                                                )                                                                                            ∂                                                  T                          2                                                                                      ⁢                    δ                    ⁢                                                                                  ⁢                                          T                      2                                                                      ]                                              ≈                                    -                              I                H                                      ⁢                          A              G                        ⁢                                                            ∂                                      (                                                                  ΔΦ                                                  D                          ⁢                                                                                                          ⁢                          1                                                                    ⁡                                              (                                                  T                          1                                                )                                                              )                                                                    ∂                                      T                    1                                                              ⁡                              [                                                      δ                    ⁢                                                                                  ⁢                                          T                      1                                                        -                                      δ                    ⁢                                                                                  ⁢                                          T                      2                                                                      ]                                                                        (        11        )            
where ∂[ΔΦD1(T1)]/∂T1≈∂[ΔΦD2(T2)]/∂T2 represents the sensitivity of the temperature sensing diodes (for current biased p/n diodes it is about −2mV/° K).
The electro-thermal coefficient “A” depends on the USSS heater power in FIG. 3 and is given by,
                    A        ≈                              -                          I              H                                ⁢                                    A              G                        ⁡                          [                                                ∂                                      (                                                                  ΔΦ                                                  D                          ⁢                                                                                                          ⁢                          1                                                                    ⁡                                              (                                                  T                          1                                                )                                                              )                                                                    ∂                                      T                    1                                                              ]                                      ≈                              -                          I              H                                ⁢                                    A              G                        ⁡                          [                                                ∂                                      (                                                                  ΔΦ                                                  D                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                                                  T                          2                                                )                                                              )                                                                    ∂                                      T                    2                                                              ]                                                          (        12        )            
The value of “A” depends on the voltage amplifier's gain and frequency response. The amplifier's voltage gain is AG=A0/(1+jω/ω0) with ω0≈104rad/sec and A0≈104. For ∂[ΔΦD2(T2)]/∂T2−2 mV/° K, and IH≈2 μA, the value of the electro-thermal coefficient is given by A≈4×10−5/(1+jω/ω0) W/° K. For proper operation, analysis shows that the electro-thermal coefficient “A” must be large relative to conductances G1* and G2*, which are about 10−7 W/° K.
The USSS (see FIG. 3) photoresponse is computed with thermal analysis of the USSS thermal equivalent circuit shown in FIG. 5. The USSS analysis follows the approach presented for conventional bolometers. For bidirectional temperature tracking, THB is always less than T1 and T2. As anticipated, thermal analysis reveals that the conductance of G1* is severely reduced by electro-thermal feedback. This reduction produces greater than a ten-fold improvement in thermal isolation and increase in photoresponse. The thermal balance conditions at the absorber 10 and intermediate stages 16 are expressed in terms of two integral/differential equations. At the absorber stage 10, the thermal balance equation is given by:
                                          Q            R                    -                      Q                          D              ⁢                                                          ⁢              1                                +                      Q                          S              ⁢                                                          ⁢              1                                      =                                            -                                                ∫                                      T                    1                                                        T                    2                                                  ⁢                                                                            G                      1                                        ⁡                                          (                      T                      )                                                        ⁢                                                                          ⁢                                      ⅆ                    T                                                                        +                                          ∫                                  T                  1                                                                      T                    1                                    +                                      δ                    ⁢                                                                                  ⁢                                          T                      1                                                                                  ⁢                              jω                ⁢                                                                  ⁢                                                      C                    1                                    ⁡                                      (                    T                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  T                                                              =                                    ∑                              n                =                0                            ∞                        ⁢                                                  ⁢                          (                                                                    -                                                                                                                                                                               ∂                              n                                                        ⁢                                                                                          G                                1                                                            ⁡                                                              (                                T                                )                                                                                                                                          ∂                                                          T                              n                                                                                                                                                                          T                        1                                                                              ⁢                                                                                    (                                                                              T                            2                                                    -                                                      T                            1                                                                          )                                                                    n                        +                        1                                                                                                            (                                                  n                          +                          1                                                )                                            !                                                                      +                                                                                                                           jω                        ⁢                                                                                                            ∂                              n                                                        ⁢                                                                                          C                                1                                                            ⁡                                                              (                                T                                )                                                                                                                                          ∂                                                          T                              n                                                                                                                                                                                        T                      1                                                        ⁢                                                                                    (                                                  δ                          ⁢                                                                                                          ⁢                                                      T                            1                                                                          )                                                                    n                        +                        1                                                                                                            (                                                  n                          +                          1                                                )                                            !                                                                                  )                                                          (        13        )            
For small temperature signals G1(T) and C1(T), the integrals in Equation (13) can be approximated by the first two terms of the Taylor series, and the temperature differential of Equation (13) yields an expression given by:GRδTS=└G1*+GD1il +jωC1*┘δT1−G1*δT2  (14)where, G1*=G1+(T2−T1)∂G1/∂T1.
Similarly, thermal balance conditions at the intermediate stage 16 give rise to a differential/integral equation given by:
                                          -                          Q                              D                ⁢                                                                  ⁢                2                                              +                      Q            H                    +                      Q                          S              ⁢                                                          ⁢              2                                      =                                            -                                                ∫                                      T                    2                                                        T                    HB                                                  ⁢                                                                            G                      2                                        ⁡                                          (                      T                      )                                                        ⁢                                                                          ⁢                                      ⅆ                    T                                    ⁢                                                            ∫                                              T                        2                                                                                              T                          2                                                +                                                  δ                          ⁢                                                                                                          ⁢                                                      T                            2                                                                                                                ⁢                                          jω                      ⁢                                                                                          ⁢                                                                        C                          2                                                ⁡                                                  (                          T                          )                                                                    ⁢                                                                                          ⁢                                              ⅆ                        T                                                                                                                  +                                          ∫                                  T                  1                                                  T                  2                                            ⁢                                                                    G                    1                                    ⁡                                      (                    T                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  T                                                              =                                                    ∑                                  n                  =                  0                                ∞                            ⁢                              [                                  (                                                                                    -                                                                                                                                                                                                           ∂                                  n                                                                ⁢                                                                                                      G                                    2                                                                    ⁡                                                                      (                                    T                                    )                                                                                                                                                              ∂                                                                  T                                  n                                                                                                                                                                                                      T                            2                                                                                              ⁢                                                                                                    (                                                                                          T                                HB                                                            -                                                              T                                2                                                                                      )                                                                                n                            +                            1                                                                                                                                (                                                          n                              +                              1                                                        )                                                    !                                                                                      +                                          jω                      ⁢                                                                                                                                                                                             ∂                                n                                                            ⁢                                                                                                C                                  2                                                                ⁡                                                                  (                                  T                                  )                                                                                                                                                    ∂                                                              T                                n                                                                                                                                                                                        T                          2                                                                    ⁢                                                                                                    (                                                          δ                              ⁢                                                                                                                          ⁢                                                              T                                2                                                                                      )                                                                                n                            +                            1                                                                                                                                (                                                          n                              +                              1                                                        )                                                    !                                                                                                      )                                ]                                      +                                          ∑                                  n                  =                  0                                ∞                            ⁢                              [                                                                                                                                                                                 ∂                            n                                                    ⁢                                                                                    G                              1                                                        ⁡                                                          (                              T                              )                                                                                                                                ∂                                                      T                            n                                                                                                                                                            T                      1                                                        ⁢                                                                                    (                                                                              T                            2                                                    -                                                      T                            1                                                                          )                                                                    n                        +                        1                                                                                                            (                                                  n                          +                          1                                                )                                            !                                                                      ]                                                                        15        )            
For small temperature signal G1(T) and C1(T) dependent weakly on temperature, the integrals are approximated by the first two Taylor series terms. Taking the temperature differential of Equation (15) and combining with Equations (11) and (12), we obtained:└G1*A┘δT1=└G1*+G2*+GD2+A+jωC2*┘δT2  (16)
Variable G2* is defined as: G2*=G2+(THB-T2)∂G2/∂T2. For sufficiently large A>>[G1*, G2*, G*D2], Equation (16) shows that electro-thermal feedback forces δT1≈δT2. This condition minimizes change in the thermal current through G: with changes in absorber's temperature T1. Having the intermediate stage 16 track the absorber's temperature effectively makes G1* behave as a much smaller thermal conductance. In the limit as A goes to infinity δT1=δT2, and the absorber's thermal isolation should approach the ideal radiative limit GD1.
Responsivity also improves because electro-thermal feedback greatly improves the effective thermal isolation. This improvement is revealed by combining Equations (14) with (16) to eliminate T2, and obtaining an expression for T1 as a function of TS, specifically:
                              δ          ⁢                                          ⁢                      T            1                          =                                                            G                R                                            [                                                      G                                          D                      ⁢                                                                                          ⁢                      1                                                        +                                      jω                    ⁢                                                                                  ⁢                                          C                      1                      *                                                        +                                                                                    G                        1                        *                                            ⁡                                              [                                                                              G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                +                                                      jω                            ⁢                                                                                                                  ⁢                                                          C                              2                              *                                                                                                      ]                                                                                    [                                                                        G                          1                          *                                                +                                                  G                          2                          *                                                +                                                  G                                                      D                            ⁢                                                                                                                  ⁢                            2                                                                          +                        A                        +                                                  jω                          ⁢                                                                                                          ⁢                                                      C                            2                                                                                              ]                                                                      ]                                      ⁢            δ            ⁢                                                  ⁢                          T              S                                ≅                                                                      G                  R                                /                                  G                                      D                    ⁢                                                                                  ⁢                    1                                                                              [                                  1                  +                                                            jω                      ⁢                                                                                          ⁢                                              C                        1                        *                                                                                    G                                              D                        ⁢                                                                                                  ⁢                        1                                                                                            ]                                      ⁢            δ            ⁢                                                  ⁢                          T              S                                                          (        17        )            
Comparing Equations (17) and (3) reveals that with large electro-thermal values A>>[G1*, G2*, GD2] the responsivity increases. The electro-thermal coefficient A≈4×10−5/(1+ω/ω0)W/° K is much larger than typical values for G1*≈G2*≈2×10−7W/° K, and the result is that electro-thermal feedback reduces conductance G1* to about GD1. This reduction represents better than a 40 fold decrease in the effective thermal conductance of GD1. The improved responsivity is limited by the optical solid angle, and, according to Equation (17), is approximated by: δT1/δTS=1/(4F2). With greatly improved photoresponsivity (X40), susceptibility to absorber stage noise and readout circuit noise are reduced thereby facilitating an improved NEΔT.
The USSS voltage photoresponse is calculated in several steps, starting by substituting Equation (10) for QH into Equation (15) and taking the differential. Under the constant bias current IH, the differential power delivered by the amplifier is simply δQH=−δVSIH. Incorporating this into the differential of Equation (15), and after some rearrangements, a new equation is obtained given by,δVSIH=G1*δT1−└G1*+G2*+GD2+jωC2┘δT2  (18)
The voltage responsivity in terms of changes in scene temperature δTS is obtained by first using Equation (16) to factor δT2 from Equation (18) and from the result factoring δT1 with Equation (17). Performing these substitutions, and after some rearrangements, the USSS's voltage responsivity is given by,
                                          δ            ⁢                                                  ⁢                                          V                S                            ⁡                              (                ω                )                                                          δ            ⁢                                                  ⁢                                          T                S                            ⁡                              (                ω                )                                                    =                                                            -                                  AG                  R                                            /                              I                H                                                                                      [                                                                                                              1                          +                                                                                    (                                                                                                G                                  1                                  *                                                                +                                A                                                            )                                                        /                                                                                                                                                                                        (                                                                                    G                              2                              *                                                        +                                                          G                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      +                                                          jω                              ⁢                                                                                                                          ⁢                                                              C                                2                                                                                                              )                                                                                                      ]                                ⁡                                  [                                                            G                                              D                        ⁢                                                                                                  ⁢                        1                                                              +                                          jω                      ⁢                                                                                          ⁢                                              C                        1                                                                              ]                                            +                              G                1                *                                              ≈                                    -                              (                                                                            G                      2                      *                                        +                                          G                                              D                        ⁢                                                                                                  ⁢                        2                                                                                                  I                    H                                                  )                                      ⁢                          (                                                G                  R                                                  G                                      D                    ⁢                                                                                  ⁢                    1                                                              )                        ⁢                                          [                                  1                  +                                      jω                    ⁢                                                                                  ⁢                                                                  C                        2                                            /                                              (                                                                              G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                                      )                                                                                            ]                                            [                                  1                  +                                      jω                    ⁢                                                                                  ⁢                                                                  C                        1                                            /                                              G                                                  D                          ⁢                                                                                                          ⁢                          1                                                                                                                    ]                                      ⁢                                                  ⁢            Volts            ⁢                          /                        ⁢            Kelvin                                              (        19        )            
The approximations to the right side of Equation (19) are obtained with A>>[G1*, G2*, GD2], and A is defined by Equation (12). The responsivity approximation equals about (0.25/F2)(G2*/IH)≈0.025V/° K: when IH=2 μA, GD1/GR=4F2=0.25 and G2*≈2×10−7 W/° K. Thus, by having made A0 very large, the absorber element's dc response greatly increases and produces a signal sufficiently large to diminish the impact of electronics noise. There are two ac time constants, the first one is from the absorber element and is given as C1/GD1, and the second is for the intermediate stage and it is given by C2/(G2*+GD1). For television frame rates, the bolometer time constant is required to be consistent with 30Hz frame rate, or C1/GD1=1/(60π). The thermal time constant is minimized by using concentrators such as microlenses or microantennas. The concentrators will reduce further the value of GD1 to provide even better responsivity. The projected sensitivity improvements with the USSS are examined next.
The USSS model includes conventional thermal noise source and the electronic/thermal noise from electro-thermal feedback. The thermal noise sources include fluctuations in: the scene's flux, QS, the radiation shields, QS1 and QS2, the heat bath coupled through conductance G1* and G2*, the absorber element, QD1, the intermediate stage, QD2, and electro-thermal feedback noise is categorized as electronic readout noise. These noise sources manifest themselves as a fluctuation in the output voltage signal VS. Specifically, noise voltage fluctuations, produced by temperature fluctuations include: (1) the scene, δVS(TS), (2) the heat bath, δVS(THB), (3) the absorber stage, δVS(T1), (4) the intermediate stage, δVS(T2), and electro-thermal/readout circuits noise voltage, δVS(EL). These noise sources are addressed next.
Spectral fluctuations in the scene temperature, δTS2(ω)/dω=4kTS2GR, contributes RMS noise to the USSS output node δVS(TS). This contribution is computed by integrating the scene fluctuations times the squared absolute value of the transfer function given by Equation (19). Approximating Equation (19) at high frequency, the RMS voltage noise due to the spectral temperature fluctuations in the scene temperature TS(ω) is:
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (                              T                S                            )                                      ≅                              [                                                            G                  2                  *                                +                                  G                                      D                    ⁢                                                                                  ⁢                    2                                                                              I                H                                      ]                    ⁢                                                    1                                  2                  ⁢                  π                                            ⁢                                                ∫                  0                  ∞                                ⁢                                                      4                    ⁢                                                                                  ⁢                                          kT                      S                      2                                        ⁢                                          G                      R                                        ⁢                                          ⅆ                      ω                                                                                                                                                                          G                                                      D                            ⁢                                                                                                                  ⁢                            1                                                                          +                                                  jω                          ⁢                                                                                                          ⁢                                                      C                            1                            *                                                                                                                                      2                                                                                      ≅                              [                                                            G                  2                  *                                +                                  G                                      D                    ⁢                                                                                  ⁢                    2                                                                              I                H                                      ]                    ⁢                                    (                                                G                  R                                                  G                                      D                    ⁢                                                                                  ⁢                    1                                                              )                                      1              /              2                                ⁢                                    (                                                kT                  S                  2                                                  C                  1                  *                                            )                                      1              /              2                                                          (        20        )            
Contributions from fluctuations in the heat bath's temperature to the output node voltage noise δVS(THB) is calculated in terms of temperature fluctuations induced on the absorber's temperature T1 and the intermediate stage's temperature T2. These relationships are obtained by taking the differentials of Equations (13) and (15) under the conditions that δTHB≠0 and δTS=0. Taking differentials of Equations (13), and (15), the following is obtained:└GS1+jωCHB┘δTHB(ω)+└G1*┘δT2(ω)=└G1*+GD1+jωC1*┘δT1(ω)  (21)└G2*+GS2+jωCHB┘δTHB(ω)+└G1*+A┘δT1(ω)=└G1*+G2*+GD2+A+jωC2*┘δT2(ω)  (22)
An expression for δVS(ω) in terms of δTHB(ω) is obtained by combining Equations (21), (22), (10), and (12).This expression is simplified with approximations that are GS2≈GS1, and GD2≈GD1=GS2≈GS1, and C2*≈C1. Combing all these yields a transfer function relating the output voltage to changes in the heat bath temperature, and it is given by,
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (              ω              )                                      =                                                            G                2                *                            +                              G                                  D                  ⁢                                                                          ⁢                  2                                                                    I              H                                ⁢                                    (                                                -                  A                                ⁢                                                                  ⁢                jω                ⁢                                                                  ⁢                                  C                  HB                                ⁢                δ                ⁢                                                                  ⁢                                                      T                    HB                                    ⁡                                      (                    ω                    )                                                              )                                      [                                                                                                                                            (                                                                                    G                              1                              *                                                        +                            A                                                    )                                                ⁢                                                  (                                                                                    G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      +                                                          jω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                                              )                                                                    +                                                                                                                                                          (                                                                              G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                +                                                      jω                            ⁢                                                                                                                  ⁢                                                          C                              2                              *                                                                                                      )                                            ⁢                                              (                                                                              G                            1                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              1                                                                                +                                                      jω                            ⁢                                                                                                                  ⁢                                                          C                              1                              *                                                                                                      )                                                                                                        ]                                                          (        23        )            
The RMS noise level is computed with the definition of the noise power spectral density, and of the transfer function given by Equation (23). Converting Equation (23) into a power spectral density, the RMS noise voltage due to thermal fluctuations in the heat bath temperature become,
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (                              T                HB                            )                                      ≅                                                            G                2                *                            +                              G                                  D                  ⁢                                                                          ⁢                  2                                                                    I              H                                ⁢                                                                      ∫                  0                  ∞                                ⁢                                                                                                                                                          [                                                                                                                            ⅆ                                  ω                                                                /                                2                                                            ⁢                              π                                                        ]                                                    ⁢                                                      ω                            2                                                    ⁢                                                      C                            HB                            2                                                    ⁢                          4                          ⁢                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              1                                                                                ⁢                                                                                    kT                              HB                              2                                                        /                                                                                                                                                                                        (                                                                                    G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                            2                                                        +                                                                                          ω                                2                                                            ⁢                                                              C                                HB                                2                                                                                                              )                                                                                                                                                                                                                        (                                                                                    G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      +                                                          j                              ⁢                                                                                                                          ⁢                              ω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                                              )                                                +                                                                                                                                                                              (                                                                                                            G                                      2                                      *                                                                        +                                                                          G                                                                              D                                        ⁢                                                                                                                                                                  ⁢                                        2                                                                                                              +                                                                          j                                      ⁢                                                                                                                                                          ⁢                                      ω                                      ⁢                                                                                                                                                          ⁢                                                                              C                                        2                                        *                                                                                                                                              )                                                                                                                                                                                                                      (                                                                                                            G                                      1                                      *                                                                        +                                                                          G                                                                              D                                        ⁢                                                                                                                                                                  ⁢                                        1                                                                                                              +                                                                          j                                      ⁢                                                                                                                                                          ⁢                                      ω                                      ⁢                                                                                                                                                          ⁢                                                                              C                                        1                                        *                                                                                                                                              )                                                                                                                                              A                                                                                                            2                                                              ⁢                                                                                  ≅                                                            G                2                *                            +                              G                                  D                  ⁢                                                                          ⁢                  2                                                                    I              H                                ⁢                                                    kT                HB                2                                            C                1                *                                                                        24        )            
We've simplified Equation (24) by recognizing that at higher frequencies the integral is simplified because GD1<<ω2C2HB, and the electro-thermal coefficient A>>[G1*+G2*+GD2]. The right side of Equation (24) is obtained by including the fact that (A+G1*+G2*+GD2)C1>>GD1C2.
The noise voltage δVS(T1) due to fluctuations in the absorber's temperature δT1(ω) is calculated using the equivalent circuit in FIG. 6, when δQ2=δQH=0. Summing the flow of all the thermal power at node T1, we obtain:
                                                                        δ                ⁢                                                                  ⁢                                                      Q                    1                                    ⁡                                      (                    ω                    )                                                              =                            ⁢                                                                    (                                                                  G                        1                        *                                            +                                              G                                                  D                          ⁢                                                                                                          ⁢                          1                                                                    +                                              j                        ⁢                                                                                                  ⁢                        ω                        ⁢                                                                                                  ⁢                                                  C                          1                          *                                                                                      )                                    ⁢                  δ                  ⁢                                                                          ⁢                                                            T                      1                                        ⁡                                          (                      ω                      )                                                                      -                                                      G                    1                    *                                    ⁢                  δ                  ⁢                                                                          ⁢                                                            T                      2                                        ⁡                                          (                      ω                      )                                                                                                                                              =                            ⁢                                                [                                                            G                                              D                        ⁢                                                                                                  ⁢                        1                                                              +                                          j                      ⁢                                                                                          ⁢                      ω                      ⁢                                                                                          ⁢                                              C                        1                        *                                                              +                                                                                            G                          1                          *                                                ⁡                                                  (                                                                                    G                              2                              *                                                        +                                                          G                                                              D                                ⁢                                                                                                                                  ⁢                                2                                                                                      +                                                          jω                              ⁢                                                                                                                          ⁢                                                              C                                2                                *                                                                                                              )                                                                                            (                                                                              G                            1                            *                                                    +                                                      G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                +                          A                          +                                                      j                            ⁢                                                                                                                  ⁢                            ω                            ⁢                                                                                                                  ⁢                                                          C                              2                              *                                                                                                      )                                                                              ]                                ⁢                δ                ⁢                                                                  ⁢                                                      T                    1                                    ⁡                                      (                    ω                    )                                                                                                          (        25        )            
δQ1 squared equals to the squared power spectral density and it is given by dQ21/df=4GD1kB(T1)2. Using Equation (16), δT2(ω) is eliminated to yield the right side of Equation (25). Combining the right side of Equation (11) with Equation (12) and using Equation (16) to eliminate δT2(ω), the spectral noise voltage δVS[T1(ω)] produced by temperature fluctuations δT1(ω) is obtained and is given by,
                                                                                          -                  δ                                ⁢                                                                  ⁢                                                      V                    S                                    ⁡                                      [                                                                  T                        1                                            ⁡                                              (                        ω                        )                                                              ]                                                  ⁢                                  I                  H                                            =                            ⁢                                                A                  ⁡                                      [                                          1                      -                                                                        δ                          ⁢                                                                                                          ⁢                                                                                    T                              2                                                        ⁡                                                          (                              ω                              )                                                                                                                                δ                          ⁢                                                                                                          ⁢                                                                                    T                              1                                                        ⁡                                                          (                              ω                              )                                                                                                                                            ]                                                  ⁢                δ                ⁢                                                                  ⁢                                                      T                    1                                    ⁡                                      (                    ω                    )                                                                                                                          =                            ⁢                              A                ⁡                                  [                                                                                    G                        2                        *                                            +                                              G                                                  D                          ⁢                                                                                                          ⁢                          2                                                                    +                                              j                        ⁢                                                                                                  ⁢                        ω                        ⁢                                                                                                  ⁢                                                  C                          2                          *                                                                                                                                    G                        1                        *                                            +                                              G                        2                        *                                            +                                              G                                                  D                          ⁢                                                                                                          ⁢                          2                                                                    +                      A                      +                                              j                        ⁢                                                                                                  ⁢                        ω                        ⁢                                                                                                  ⁢                                                  C                          2                          *                                                                                                      ]                                                                                                                      ⁢                              δ                ⁢                                                                  ⁢                                                      T                    1                                    ⁡                                      (                    ω                    )                                                                                                          (        26        )            
Solving Equation (25) for δT1(ω) and using it to eliminate δT1(107 ) from Equation (26), an analytical solution is obtained relating the spectral noise voltage δVS[T1(107 )] to fluctuations in the noise power δQ1. Squaring the absolute value of this solution, replacing δQ1 by dQ21/df=4GD1kB(T1)2, integrating [0≦ω≦∞], and taking the square root, the RMS voltage fluctuations δVS(T1), produced by δT1(ω), is obtained and it is given by:
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (                              T                1                            )                                      =                                            1                              I                H                                      ⁢                                                                                ∫                    0                    ∞                                    ⁢                                                            2                      ⁢                                              AG                                                  D                          ⁢                                                                                                          ⁢                          1                                                                    ⁢                                              k                        B                                            ⁢                                              T                        1                        2                                            ⁢                                                                        ⅆ                          ω                                                /                        π                                                                                                                                                                                                                                                                                          G                                  1                                  *                                                                +                                                                  G                                                                      D                                    ⁢                                                                                                                                                  ⁢                                    1                                                                                                  +                                                                  j                                  ⁢                                                                                                                                          ⁢                                  ω                                  ⁢                                                                                                                                          ⁢                                                                      C                                    1                                    *                                                                                                  +                                                                                                                                                                                                                                          [                                                                                                                                                    (                                                                                  1                                          +                                          A                                                                                )                                                                            ⁢                                                                              G                                                                                  D                                          ⁢                                                                                                                                                                          ⁢                                          1                                                                                                                                                      +                                                                          j                                      ⁢                                                                                                                                                          ⁢                                      ω                                      ⁢                                                                                                                                                          ⁢                                                                              C                                        1                                        *                                                                                                                                              ]                                                                /                                                                  [                                                                                                            G                                      2                                      *                                                                        +                                                                          G                                                                              D                                        ⁢                                                                                                                                                                  ⁢                                        2                                                                                                              +                                                                          j                                      ⁢                                                                                                                                                          ⁢                                      ω                                      ⁢                                                                                                                                                          ⁢                                                                              C                                        2                                        *                                                                                                                                              ]                                                                                                                                                                                                                2                                                                      ⁢                                                                                                =                                                                                          G                    2                    *                                    +                                      G                                          D                      ⁢                                                                                          ⁢                      2                                                                                        I                  H                                            ⁡                              [                                                                            k                      B                                        ⁢                                          T                      1                      2                                                                            C                    1                    *                                                  ]                                                    1              /              2                                                          (        27        )            
Temperature fluctuations in the intermediate stage δT2(ω) produce noise voltage at the output node and this contribution is calculated as contributions from the absorber stage δT1(107 ). Using the equivalent circuit in FIG. 6, the power at node T1 is summed when δQ1=δQH=0, and a relationship between δT1(ω) and δT2(ω) is obtained and given by:δT1(ω)=└G1*/(G1*+GD2+jωC1*)┘δT2(ω)  (28)
The intermediate stage noise source δQ2, produces temperature fluctuations δT2(ω), hence |T2|>|T1|, and this is given by Equation (28). Summing the power at node T2 in FIG. 6, a spectral power relationship given by the following is obtained:δQ2(ω)+δQH(ω)=└G1*+G2*+GD2+jωC2*┘δT2(ω)−G1*δT1(ω)  (29)
The variables δQHand δT1 in Equation (29) are eliminated with δQH=A[δT1-δT2], and Equation (28), respectively, to obtain:δQ2(ω)=└(G2*+GD2+jωC2*)+(A+G1*)(GD1+jωC1*)/G1*+GD1+jωC1*)┘δT2(107)   (30)
Using Equations (11) and (28), an expression is obtained for the spectral fluctuations in the output voltage VS[T2(ω)] produced by thermal fluctuations from the intermediate stage and it is given by:−δVS[T2(ω)]IH=A[1−δT1(ω)]δT2(ω)=−A(GD1+jωC1*)(G1*+GD1+jωC1*)δT2(ω)  (31)
Fluctuations δQ2 squared is given in terms of black body radiation by dQ22/df=4GD2kB(T2)2. Combining Equations (30) and (31), to eliminate δT2(ω), and taking the absolute value, squared, integrating [0≦ω≦∞], and taking the square root yields the voltage fluctuations at node T2 in terms of the intermediate stage's temperature given by,
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (                              T                2                            )                                      =                                            1                              I                H                                      ⁢                                                                                ∫                    0                    ∞                                    ⁢                                                            2                      ⁢                                              G                                                  D                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              k                        B                                            ⁢                                              T                        2                        2                                            ⁢                                                                        ⅆ                          ω                                                /                        π                                                                                                                                                              1                          +                                                                                    G                              1                              *                                                        A                                                    +                                                                                                                                                                                          (                                                                                                                  G                                        2                                        *                                                                            +                                                                              G                                                                                  D                                          ⁢                                                                                                                                                                          ⁢                                          2                                                                                                                    +                                                                              j                                        ⁢                                                                                                                                                                  ⁢                                        ω                                        ⁢                                                                                                                                                                  ⁢                                                                                  C                                          2                                          *                                                                                                                                                      )                                                                                                                                                                                                                                    (                                                                                                                  G                                        1                                        *                                                                            +                                                                              G                                                                                  D                                          ⁢                                                                                                                                                                          ⁢                                          1                                                                                                                    +                                                                              j                                        ⁢                                                                                                                                                                  ⁢                                        ω                                        ⁢                                                                                                                                                                  ⁢                                                                                  C                                          1                                          *                                                                                                                                                      )                                                                                                                                                                                      A                              ⁡                                                              (                                                                                                      G                                                                          D                                      ⁢                                                                                                                                                          ⁢                                      1                                                                                                        +                                                                      j                                    ⁢                                                                                                                                                  ⁢                                    ω                                    ⁢                                                                                                                                                  ⁢                                                                          C                                      1                                      *                                                                                                                                      )                                                                                                                                                                                2                                                                      ⁢                                                                                                =                                                                      G                  2                  *                                +                                  G                                      D                    ⁢                                                                                  ⁢                    2                                                                              I                H                                      ⁡                          [                                                                    k                    B                                    ⁢                                      T                    2                    2                                                                    C                  2                  *                                            ]                                                          (        32        )            
Equation (32) is simplified by recognizing that for large “A” the approximation on the right is obtained and it reveals that the voltage fluctuations are proportional to the classical temperature fluctuations of a black body at temperature T2. The factor in front is the total thermal conductivity from the intermediate stage divided by the amplifier's bias current IH. The total fluctuation in voltage VS from fluctuations in temperatures is the RMS sum of fluctuations from THB, TS, T1 and T2.
Electrical noise from the R/O and electro-thermal feedback circuit corrupts the USSS output signal. The level of corruption has been previously computed, and is given in terms of the amplifier's noise current, the conductivity of the thermal bridges, and amplifier's bias current IH. The contribution of electrical noise took into account the affects of electro-thermal feedback and electrical feedback. In the analysis below, the amplifier's effective channel noise IN0 without any feedback is defined. Noise analysis is performed by representing noise as a Fourier series sum, and analyzing the contribution of each term. Including electrical and thermal feedback, the electrical voltage noise is given by,
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          [                                                E                  L                                ⁡                                  (                  ω                  )                                            ]                                      =                                            Z              S                        ⁢                                          I                                  N                  ⁢                                                                          ⁢                  0                                            ⁡                              (                ω                )                                                          1            +                                                            g                  m                                ⁢                                  Z                  S                                ⁢                                  Z                  D                                                            (                                                      Z                    S                                    +                                      Z                    G                                    +                                      Z                    D                                                  )                                      -                          [                                                                                                                  [                                                                                                            g                              m                                                        ⁢                                                          Z                              S                                                        ⁢                                                          Z                              G                                                                                                                                          Z                              S                                                        +                                                          Z                              G                                                        +                                                          Z                              D                                                                                                      ]                                                                                                                                                                          I                          H                                                ⁢                                                                              ∂                                                                                          E                                F                                                            ⁡                                                              (                                T                                )                                                                                                                                          ∂                            T                                                                          ⁢                                                  (                                                                                    G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      +                                                          j                              ⁢                                                                                                                          ⁢                              ω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                                              )                                                                                                                                                                                                        (                                                                              G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                +                                                      j                            ⁢                                                                                                                  ⁢                            ω                            ⁢                                                                                                                  ⁢                                                          C                              2                              *                                                                                                      )                                                                                                                                                                          (                                                                                    G                              1                              *                                                        +                                                          G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      +                                                          j                              ⁢                                                                                                                          ⁢                              ω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                                              )                                                +                                                                                                                                                                          G                          1                          *                                                ⁡                                                  (                                                                                    G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                      +                                                          j                              ⁢                                                                                                                          ⁢                              ω                              ⁢                                                                                                                          ⁢                                                              C                                1                                *                                                                                                              )                                                                                                                                ]                                                          (        33        )            
Affects of electrical feedback are represented by the first algebraic term in denominator, and affects of thermal feedback are represented by the second algebraic term in the denominator. The larger the denominator is the smaller is the noise. The negative sign in front of the thermal feedback term is cancelled when it is multiplied by the negative factor ∂EF(T)/∂T≈2 mV/° K. Noise attenuation produced by the electrical and thermal feedbacks terms is estimated in terms of thermal conductivities. Typically, ZG>>[ZD, ZS], gmZS>>1, G2*≈G1*>[10 GD1, 10 GD2]. Additionally, the USSS operating conditions are given by: IH=2 μA, ∂EF(T)/∂T≈2 mV/° K, [GD1, GD2]≈1.5×10−8W/° K, gmZS≈104, and [G1*, G2*]≈2×10−7 W/° K. The value of the electro-thermal term A≈4×10−5W/° K dominates over the other thermal conductivities. Incorporating all these into Equation (33), a simplified expression is obtained, and it is given as,
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          [                                                E                  L                                ⁡                                  (                  ω                  )                                            ]                                      ≅                                            I                              N                ⁢                                                                  ⁢                0                                      ⁡                          (              ω              )                                ⁢                                                    Z                S                            [                              1                +                                                                            -                                              g                        m                                                              ⁢                                          Z                      S                                        ⁢                                          I                      H                                        ⁢                                                                  ∂                                                                              E                            F                                                    ⁡                                                      (                            T                            )                                                                                                                      ∂                        T                                                              ⁢                                          (                                                                        G                                                      D                            ⁢                                                                                                                  ⁢                            1                                                                          +                                                  j                          ⁢                                                                                                          ⁢                          ω                          ⁢                                                                                                          ⁢                                                      C                            1                            *                                                                                              )                                                                                                  (                                                                        G                          2                          *                                                +                                                  G                                                      D                            ⁢                                                                                                                  ⁢                            2                                                                          +                                                  j                          ⁢                                                                                                          ⁢                          ω                          ⁢                                                                                                          ⁢                                                      C                            2                            *                                                                                              )                                        ⁢                                          (                                                                        G                          1                          *                                                +                                                  G                                                      D                            ⁢                                                                                                                  ⁢                            1                                                                          +                                                  j                          ⁢                                                                                                          ⁢                          ω                          ⁢                                                                                                          ⁢                                                      C                            1                            *                                                                                              )                                                                                  ]                                      -              1                                      ≅                                            I                              N                ⁢                                                                  ⁢                0                                      ⁡                          (              ω              )                                ⁢                                    Z              S                        ⁡                          [                                                                    G                    2                    *                                    ⁢                                      G                    1                    *                                                                    AG                                      D                    ⁢                                                                                  ⁢                    1                                                              ]                                                          (        34        )            
The right side of Equation (34) incorporates these approximations combined with A=−IHgmZS∂EF(T)/∂T, given by Equation (12). For large electro-thermal feedback (A>>[G1*, G2*, and GD1] the noise is attenuated. The spectral noise density is readily converted into a standard deviation by integrating over frequency. If 1/f noise is neglected, the integral of Equation (34) is simplified and the noise voltage, in terms of the electrical bandwidth, (ΔBW)1/2, is given by:
                              δ          ⁢                                          ⁢                                    V              S                        ⁡                          (                              E                L                            )                                      ≤                                            I                              N                ⁢                                                                  ⁢                0                                      ⁡                          (              ω              )                                ⁢                                    Z              S                        ⁡                          [                                                                    G                    2                                    ⁢                                      G                    1                                                                    AG                                      D                    ⁢                                                                                  ⁢                    1                                                              ]                                ⁢                      Δ            BW                          1              /              2                                                          (        35        )            
The total noise at the USSS output node is the RMS sum of contributions from: the scene [Equation (20)], the heat bath [Equation (24)], the bolometer stage T1 [Equation (27)], the intermediate stage T2 [Equation (32)], and the readout electronics [Equation (35)]. The USSS sensitivity, NEΔTUSSS, or minimum resolvable temperature, is obtained by equating the total noise to the signal: unity signals to noise ratio. RMS summing Equations (20), (24), (27), (32), (35) and dividing it by the absolute value of right side of Equation (19) yields the expression for NEΔTUSSS, and it is given by,
                              NE          ⁢                                          ⁢          Δ          ⁢                                          ⁢                      T            USSB                          ≥                                                                                                                                                k                        B                                            ⁢                                              T                        S                        2                                                                                    C                      1                      *                                                                      ⁡                                  [                                                                                                                                                                        G                                                              D                                ⁢                                                                                                                                  ⁢                                1                                                                                                                    G                              R                                                                                +                                                                                    (                                                                                                G                                                                      D                                    ⁢                                                                                                                                                  ⁢                                    1                                                                                                                                    G                                  R                                                                                            )                                                        2                                                                                                                                                                                                                    (                                                                                                                                                                                                                                              (                                                                                                                              T                                            HB                                                                                                                                T                                            S                                                                                                                          )                                                                            2                                                                        +                                                                                                                  (                                                                                                                              T                                            1                                                                                                                                T                                            S                                                                                                                          )                                                                            2                                                                        +                                                                                                                                                                                                                                                                                                                  (                                                                                                                              T                                            2                                                                                                                                T                                            S                                                                                                                          )                                                                            2                                                                        ⁢                                                                          (                                                                                                                        C                                          1                                          *                                                                                                                          C                                          2                                          *                                                                                                                    )                                                                                                                                                                                            )                                                    +                                                                                                                                                                                                                        (                                                                                                G                                  1                                  *                                                                                                  G                                  R                                                                                            )                                                        2                                                    ⁢                                                      (                                                                                          C                                1                                *                                                                                                                              k                                  B                                                                ⁢                                                                  T                                  S                                  2                                                                                                                      )                                                    ⁢                                                                                    (                                                                                                                                    I                                                                          N                                      ⁢                                                                                                                                                          ⁢                                      0                                                                                                        ⁢                                                                      Z                                    S                                                                    ⁢                                                                      I                                    H                                                                                                  A                                                            )                                                        2                                                    ⁢                                                      Δ                            BW                                                                                                                                ]                                                            1                /                2                                      ⁡                          [                                                1                  +                                                            (                                                                        ω                          ⁢                                                                                                          ⁢                                                      C                            1                            *                                                                                                    G                                                      D                            ⁢                                                                                                                  ⁢                            1                                                                                              )                                        2                                                                    1                  +                                                            (                                                                        ω                          ⁢                                                                                                          ⁢                                                      C                            2                            *                                                                                                                                G                            2                            *                                                    +                                                      G                                                          D                              ⁢                                                                                                                          ⁢                              2                                                                                                                          )                                        2                                                              ]                                            1            /            2                                              (        36        )            
The NEΔTUSSS sensitivity in Equation (36) has been factored into three factors for comparison with the NEΔT sensitivity of conventional bolometers given by Equation (9). Sensitivity comparisons are made on a term by terms basis.
The first factor (kTS2/C1*)1/2 represents the theoretical thermal sensitivity limit for a body at temperature TS and heat capacity C1*. As expected, this factor is the same in Equations (9) and (36).
The second factor in the square brackets in Equation (36) is a sum of five terms with the upper terms corresponding to the first term in the square brackets in Equation (9). These terms show how sensitivity is reduced by thermal loading. With active thermal isolation, thermal loading is the USSS is much smaller than thermal loading in conventional bolometers: [GD1/GR]2<<[(GD1+G2*)/GR]2. This reduction in thermal loading and this leads to greater than 10X sensitivity improvement. The other thermal noise terms on the first line inside the square brackets in Equation (36) include radiation shield noise, absorber noise, and heat bath noise. These noise sources are multiplied by the factors, respectively, [GD1/GR]2 and [(GD1+G2*)/GR]2 in Equations (36) and (9). Clearly, thermal loading amplifies the affects of thermal noise from the absorber and radiation shields, and the multiplication is much worse (at least 10X) in conventional bolometers.
This affective noise “amplification” occurs because thermal loading attenuates the signal and not the thermal noise. In the NEΔT equations this attenuation manifests itself as though the noise increases. This “noise amplification” also applies to the electronics' readout noise and is apparent from comparing the second line inside the square brackets in Equation (36) with the last term inside the square brackets in Equation (9). The electronics' readout noise is smaller in the USSS compared to electronics' readout noise in conventional bolometers. This is readily confirmed by comparing the respective expressions for the readout noise in Equations (36) and (9). Comparison is facilitated by recognizing the relationship between the variable in Equations (36) and (9). The electro-thermal feedback term is given by A=IHZSgM∂EF/∂T, with ∂EF/∂T≈−2 mV/° K, and (IN0/gM)2=dEN2/df. Also, the thermal conductance G2*≈G1* and these are much larger GD1 or GR. The resistive bolometer's temperature coefficient ∂RCB/∂TS≈200, and the bias current is about ICR≈0.1 mA. Because the conventional bias current is large, it is pulsed and for a 400×500 element array, the value of ωEL≈400×500 ΔBW. Inserting all these values reveals that for the same readout transistors the impact of the readout noise in the USSS is about 50 times smaller than in a conventional bolometer. The reduced impact of readout noise in the USSS over conventional bolometers indicates that the signal in the USSS is higher thereby making the readout noise less significant.
The third factor in Equation (36) represents the AC frequency response and is given as a ratio of a zero and pole. In conventional bolometers the AC factor in Equation (9) includes only a pole at ωTM. This pole is the bolometer's thermal time constant and pulsed readout makes this factor significantly greater than one and adversely degrades sensitivity. In the USSS the AC factor includes a pole and a zero. The pole is the absorber's time constant and the zero is the intermediate stage's time constant. Since pulsed readout is not used, the ratio of these as a function of frequency can be made equal to about one.
For an absorber element 10 heat capacity C1* equal to about C1*≅1.5×10−1J/° K, the thermodynamic sensitivity limit is about (kTS2/C1*)1/2≈0.1 m° K. This is much more sensitive than the performance of conventional bolometers. The sensitivity degradation are due to thermal loading that decrease this by about 50X to about 5 m° K. Readout noise and bandwidth further reduces the sensitivity by about 4X, resulting in an NEΔT≈20 m° K. As a result, there is a need for improved sensitivity in conventional bolometers.